Please help me understand this? Expectation of f(X) proof.

Question

I don't really understand what $Im(X)$ is. Also, I don't get the first line involving summation. Why did it change to two summation signs? Could someone please explain what all the symbols beneath summation signs mean?

Thanks, And Merry Christmas!

Answer 1

$\operatorname{Im} X$ is the image of $X$, that is the set of values that the random variable $X$ can attain.

$f$ is a function that takes every value in $\operatorname{Im} X$ to a value $y=f(x)$. In the first line of the proof you posted, the elements of $\operatorname{Im} X$ are grouped according to the value $y$ they're taken to by the function $f$. In the first sum, you sum over the set $A$ of possible values $y$ that the function $f$ can assume, in the second sum you sum over those $x$ that realise a particular value $y$.